Machine learning systems and methods for training with noisy labels

ABSTRACT

Machine learning classification models which are robust against label noise are provided. Noise may be modelled explicitly by modelling “label flips”, where incorrect binary labels are “flipped” relative to their ground truth value. Distributions of label flips may be modelled as prior and posterior distributions in a flexible architecture for machine learning systems. An arbitrary classification model may be provided within the system. The classification model is made more robust to label noise by operation of the prior and posterior distributions. Particular prior and approximating posterior distributions are disclosed.

FIELD

This disclosure generally relates to machine learning, and in particular to machine learning systems and methods which are robust against label noise in training datasets in at least some circumstances.

BACKGROUND

In machine learning applications, the performance of a classification model is typically highly dependent on the availability of a large collection of high quality annotated data to use in training the model. Deep neural networks, for example, tend to be particularly sensitive to the quality of annotations. Classification models generally require large quantities of data for training. However, in most circumstances, obtaining large quantities of noise-free training data is a costly and tedious process. For instance, training data for classification models is often generated via a labeling process which annotates the training data with labels. A typical labelling process involves training multiple annotators to assign labels correctly (i.e. with a low incidence of noise) and aggregating their feedback, which tends to be resource-intensive.

It is sometimes possible to obtain lower-quality training data with relatively less effort. For instance, training data with noisy labels may be drawn from sources such as social media websites, search engines, or crowd annotation services (e.g. via the Internet). As an example, images returned from a search engine for a particular set of queries (such as cats, dogs, rabbits, etc.) may be labelled with those queries and used as training data. Some of the images returned will not feature the queried entity (e.g. results for the query “cat” may include an image for a toy mouse taken from a page relating to the care of cats), and so the applied label may be incorrect. Accordingly, the training data may be considered to have noisy labels.

It is sometimes possible to attempt to “clean up” the noisy data, for example by employing a crowdsourcing service to find incorrect labels. This may incur additional cost, but may increase the quality of the training data. It may also be possible to ignore the presence of noise and train conventional models on the noise-laden data. This may result in a loss in performance, e.g. in the form of an increased error rate relative to models with less-noisy data, but for some applications the resulting model may be “good enough” even with the effects of noise.

Some approaches, however, have attempted to provide classification models which model label noise explicitly in order to gain robustness against such noise. For instance, a technique which models confusion between labels is described by Giorgio Patrini, Alessandro Rozza, Aditya Menon, Richard Nock, and Lizhen Qu, Making neural networks robust to label noise: a loss correction approach, arXiv:1609.03683, 2016. Another technique is described by Tong Xiao, Tian Xia, Yi Yang, Chang Huang, and Xiaogang Wang, Learning from massive noisy labeled data for image classification, In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2015 and extended by Ishan Misra, C. Lawrence Zitnick, Margaret Mitchell, and Ross Girshick, Seeing through the human reporting bias: Visual classifiers from noisy human-centric labels, In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2016 to model “visual presence” and “relevance” of concepts for each instance as a way to correct for label noise introduced by human-centric biases.

There is thus a general desire for systems and methods for performing machine learning which are robust to noisy labels and are applicable to a broad range of machine learning problems.

The foregoing examples of the related art and limitations related thereto are intended to be illustrative and not exclusive. Other limitations of the related art will become apparent to those of skill in the art upon a reading of the specification and a study of the drawings.

BRIEF SUMMARY

There exists a need to be able to processor at least some problems having size and/or connectivity greater than (and/or at least not fully provided by) the working graph of an analog processor. Computational systems and methods are described which, at least in some implementations, allow for the computation of at least some problem graphs which have representations which do not fit within the working graph of an analog processor (e.g. because they require more computation devices and/or more/other couplers than the processor provides).

Some implementations provide a method for instantiating a machine learning system for classifying one or more observed elements of an input dataset, the input dataset labelled with one or more observed labels drawn from a plurality of binary labels, the method executed by at least one processor in communication with at least one memory and comprising: instantiating a recognition system in the at least one memory, the recognition system operable to classify elements of the input dataset by generating a classification probability for the at least one observed element; instantiating an approximating posterior system in the at least one memory, the approximating posterior system approximating a true posterior distribution corresponding to a prior distribution which models label noise as a set of stochastic label flips on the plurality of binary labels, each label flip indicating a belief in the correctness of a corresponding label, the approximating posterior system operable to generate a posterior probability for one or more label flips for the observed element given the one or more observed labels; and training at least one of the recognition system and the approximating posterior distribution based on a training dataset, the prior distribution, and the approximating posterior distribution.

In some implementations, the method further comprises instantiating a shared transformation system in the at least one memory, the shared transformation system operable to receive the one or more observed elements and to generate a representation of the observed elements based on one or more shared parameters, wherein the recognition system and the approximating posterior system are operable to receive the representation of the observed elements as input.

In some implementations, training at least one of the recognition system and the approximating posterior system comprises training the shared transformation system to generate the one or more shared parameters. In some implementations, at least one of the recognition system and the approximating posterior system comprises a deep neural network. In some implementations, at least one of selecting the prior distribution, selecting the recognition model, and selecting the approximating posterior distribution comprises selecting a predetermined distribution or model.

In some implementations, the prior distribution comprises a factorial distribution, a Boltzmann distribution, and/or a spike (the spike defining a high-probability state in the prior distribution). In some implementations, the high-probability state comprises the state where no labels are flipped.

In some implementations, the approximating posterior distribution comprises a factorial distribution and/or a spike (the spike defining a high-probability state in the prior distribution). In some implementations, the high-probability state comprises the state where no labels are flipped.

In some implementations, the approximating posterior distribution comprises a directed graphical model, the directed graphical model operable to generate, given an initial class, a probability of label change for each of one or more remaining classes.

In some implementations, the method further comprises selecting one of the prior distribution and the approximating posterior distribution based on a previous selection of the other one of the prior distribution and the approximating posterior distribution. In some implementations, the prior distribution comprises a Boltzmann distribution with a spike and the approximating posterior distribution comprises a factorial distribution with a spike.

In some implementations, training comprises optimizing an objective function, the objective function comprising a variational lower bound. In some implementations, the variational lower bound comprises a difference between and a true log likelihood of the data and a KL distance between the true and approximating posterior distributions. In some implementations, training comprises optimizing the variational lower bound using stochastic gradient descent.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING(S)

In the drawings, identical reference numbers identify similar elements or acts. The sizes and relative positions of elements in the drawings are not necessarily drawn to scale. For example, the shapes of various elements and angles are not necessarily drawn to scale, and some of these elements may be arbitrarily enlarged and positioned to improve drawing legibility. Further, the particular shapes of the elements as drawn, are not necessarily intended to convey any information regarding the actual shape of the particular elements, and may have been solely selected for ease of recognition in the drawings.

FIG. 1 is a flowchart illustrating an example method for instantiating a machine learning system which is robust against label noise in at least some circumstances.

FIG. 2 is a Bayesian network diagram illustrating an example set of relationships between variables in a marginalized distribution.

FIG. 3A shows schematically an architecture of an example machine learning system wherein a recognition system and an approximating posterior distribution each receive an input variable.

FIG. 3B shows schematically an architecture of an example machine learning system wherein a shared transformation system for transforming an input variable is shared between a recognition system and an approximating posterior distribution.

FIG. 4A is a schematic diagram via a directed graphical model according to a conventional approach employing directed edges.

FIG. 4B is a schematic diagram of a undirected graphical model according to at least one of the approaches according to at least one aspect of the present disclosure.

FIG. 4C is a schematic diagram of a undirected graphical model according to at least one of the approaches according to at least one aspect of the present disclosure.

FIG. 5 is a flowchart illustrating an example method for instantiating an example machine learning system according to an aspect of the present disclosure.

DETAILED DESCRIPTION

In the following description, certain specific details are set forth in order to provide a thorough understanding of various disclosed implementations. However, one skilled in the relevant art will recognize that implementations may be practiced without one or more of these specific details, or with other methods, components, materials, etc. In other instances, well-known structures associated with computer systems, server computers, and/or communications networks have not been shown or described in detail to avoid unnecessarily obscuring descriptions of the implementations.

Unless the context requires otherwise, throughout the specification and claims that follow, the word “comprising” is synonymous with “including,” and is inclusive or open-ended (i.e., does not exclude additional, unrecited elements or method acts).

Reference throughout this specification to “one implementation” or “an implementation” means that a particular feature, structure or characteristic described in connection with the implementation is included in at least one implementation. Thus, the appearances of the phrases “in one implementation” or “in an implementation” in various places throughout this specification are not necessarily all referring to the same implementation. Furthermore, the particular features, structures, or characteristics may be combined in any suitable manner in one or more implementations.

As used in this specification and the appended claims, the singular forms “a,” “an,” and “the” include plural referents unless the context clearly dictates otherwise. It should also be noted that the term “or” is generally employed in its sense including “and/or” unless the context clearly dictates otherwise.

The headings and Abstract of the Disclosure provided herein are for convenience only and do not interpret the scope or meaning of the implementations.

Aspects of the present disclosure relate to machine learning classification models which are robust against label noise in at least some circumstances. Some implementations comprise systems and methods for instantiating machine learning systems which model noise explicitly. For example, in a machine learning system with binary labels, incorrect labels may be corrected by “flipping” the labels to their correct state. A distribution of label flips may be used to model labelling errors. As another example, interactions between “true” and “noisy” labels may be represented by an undirected graphical model. Training such a model may be assisted by incorporating an auxiliary distribution to guide inference of true labels. In some implementations, a general architecture for such machine learning systems is provided so that arbitrary machine learning recognition systems may be incorporated into the architecture, thereby permitting broad applicability of the presently-disclosed systems and methods.

Modelling Label Flips

FIG. 1 shows an example method 100 for instantiating an example machine learning system according to the present disclosure. At 110, a prior distribution for modelling label noise is selected. The prior distribution models label noise explicitly. In some implementations, the prior distribution models incorrect label choices. For example, the prior distribution may model label noise in the form of stochastic “label flips” on binary labels (i.e. labels which are either applied or not). In some implementations, the prior distribution models label flips in a structured way—for instance, images labeled with “boat” may be more likely to be labeled with “water”, “kitten” with “cat”, “horse” with “animal”, and so on.

The following discussion will be assisted with the definition of some notation. The prior distribution is denoted herein by p_(θ)(t), where θ is a set of parameters and t is a vector of binary label flips. (The prior distribution may be defined over additional variables, which are omitted from the notation for simplicity.) Since p_(θ)(t) models label noise, it may generate higher probability for vectors t that are likely to correspond to label noise. Similarly, p_(θ)(t_(l)) denotes the probability of a label flip for the label l. A dataset of N elements to be processed by a machine learning system classifying L labels may be represented by {(x^((n)), y^((n)))}_(n=1) ^(N) where x^((n)) is an element of the dataset (e.g. an image and/or some other data) and y^((n))={y₁ ^((n)), y₂ ^((n)), . . . , y_(L) ^((n))} ∈ {0,1}^(L) is the multi-label annotation vector for x^((n)).

In this notation, y_(l) ^((n))=1 denotes that element x^((n)) has been labelled with label l and t_(l)=1 denotes that label l has been flipped (i.e. y_(l) ^((n)) is 0 when it should be 1 or vice-versa). For example, label flip vector t may be defined according to the following formula:

$y_{l} = \left\{ \begin{matrix} {\hat{y}}_{l} & {{{if}\mspace{14mu} t_{l}} = 0} \\ {1 - {\hat{y}}_{l}} & {{otherwise}\mspace{14mu}{\forall{l \in \left\{ {1,2,\ldots\mspace{11mu},L} \right\}}}} \end{matrix} \right.$

where ŷ^((n)) are the ground truth labels for x^((n)) (which are not necessarily the same as observed labels y^((n)) and may not be known).

Several choices are possible for the prior distribution p_(θ)(t). As is discussed elsewhere here, the selection of the prior distribution may have ramifications on the selection of the (approximating) posterior distribution (selected at 114), and vice-versa. For instance, training the resulting system may involve calculating the KL divergence between the posterior and prior distributions, a calculation which will vary in complexity depending on the pair of distributions selected. Examples of distributions which may be used include a factorial distribution, a Boltzmann distribution, and a distribution (such as a Boltzmann distribution) with a “spike”. These examples are discussed in greater detail below.

At 112, a recognition model is selected. The recognition model may be represented by p_(θ)(ŷ_(l)|x). The recognition model accepts an instance x and returns a probability of each class (each of which corresponds to a label) present in the instance, optionally along with other inputs and outputs. A computing system may instantiate the recognition model as, for example, a deep neural network, a logistic regression, and/or any other suitable machine learning technique.

At 114, an approximating posterior distribution is selected. The approximating posterior distribution approximates the true posterior distribution p_(θ)(t|x, y) and may be denoted q_(ϕ)(t|x, y), where ϕ represents the parameters of the approximating posterior distribution. Several choices are possible for the approximating posterior distribution q_(ϕ)(t|x, y). Examples of distributions which may be used include a factorial distribution, a directed graphical model, and/or a distribution with a spike. These examples are discussed in greater detail below.

One or more of the selections at 110, 112, and 114 may be made by a user and provided to a computing system. Alternatively, or in addition, one or more of the selections at 110, 112, and 114 may be made by the computing system, e.g. in response to identifying a characteristic of a problem, in response to an instruction, and/or by predetermination (e.g. the computing system may only be provided with one option to select from).

At 116 and 118, the recognition model and approximating posterior distribution are instantiated by a computing system as a recognition system and an approximating posterior system, respectively. Instantiation may comprise representing the relevant models and distributions with logical structures in a memory of the computing system. The instantiated systems direct the operation of one or more processors of the computing system. For example, the instantiated systems may comprise deep neural networks which are operable to take the inputs required by the model (or distribution) and to generate outputs which correspond to the outputs of the model (or distribution). Examples of instantiations of acts 116 and 118 are discussed in greater detail below with reference to FIGS. 3A and 3B.

At 130, the machine learning system is instantiated based on the selected distributions and trained. The prior distribution, recognition model, and posterior distribution may be trained separately or together. Their training acts are shown as 122, 124, and 126, respectively. In some implementations, the prior distribution is fixed during training; it may be trained prior to the training of the machine learning system, may be pre-determined, and/or may be fixed during training of the machine learning system. Accordingly, 122 (corresponding to the training of the prior distribution) is shown as optional by rendering it in dashed lines in FIG. 1 . In some implementations, the recognition model and posterior distribution are trained together.

As will be familiar to skilled readers, learning parameters of a machine learning system, such as a neural network, may involve optimizing a log likelihood function with the following general form:

$\max\limits_{\theta}{\frac{1}{N}{\sum\limits_{n}^{N}\;{\sum\limits_{l}^{L}\;{{\log\left( {p_{\theta}\left( {y_{l}^{(n)}❘x^{(n)}} \right)} \right)}.}}}}$

The variable t_(l) is latent, which allows the probability distribution p_(θ)(y_(l) ^((n))|x^((n))) to be decomposed by marginalizing t_(l), thereby yielding the following marginalized expression (the (n) superscript notation is omitted for simplicity):

${p_{\theta}\left( {y_{l}❘x} \right)} = {\sum\limits_{t_{l}}\;{{p_{\theta}\left( {{y_{l}❘t_{l}},x} \right)}{p_{\theta}\left( t_{l} \right)}}}$

This representation invokes the conditional distribution p_(θ)(y_(l)|t_(l), x), which may be given by: p _(θ)(y _(l) |t _(l) ,x)=p _(θ)(ŷ _(l) =y _(l) |x)^(1−t) ^(l) (1−p _(θ)(ŷ _(l) =y _(l) |x))^(t) ^(l)

where the binary flip vector t and its elements t_(l) are as defined previously.

Label flips may be structured if an appropriate prior distribution is selected. For instance, changing an image's label from “cat” to “dog” may involve changing two binary labels together. Appropriate prior distributions include certain non-factorial distributions (such as non-factorial Boltzmann distributions, with or without a spike), which may be able to represent all label flips together. In circumstances where it is necessary or desirable to represent structured label flips, however, it may be appropriate to extend the marginalized distribution given above across all label flips (i.e. to extend the above expressions to the vector t), thereby enabling the representation of correlations between individual flips t_(l). This extension may yield the following formula:

${p_{\theta}\left( {y❘x} \right)} = {\sum\limits_{t}\;{{p_{\theta}\left( {{y❘t},x} \right)}{{p_{\theta}(t)}.}}}$

This marginalized distribution may be represented using Bayesian network 200 shown in FIG. 2 . In the illustrated graphical model, t (shown as variable 204) is independent from x (shown as variable 202) unless y (shown as variable 206) is observed. Similarly, we can extend the conditional distribution given above across all label flips to yield:

${p_{\theta}\left( {{y❘t},x} \right)} = {\prod\limits_{l}\;{{p_{\theta}\left( {{\hat{y}}_{l} = {y_{l}❘x}} \right)}^{1 - t_{l}}{\left( {1 - {p_{\theta}\left( {{\hat{y}}_{l} = {y_{l}❘x}} \right)}} \right)^{t_{l}}.}}}$

In some implementations, act 130 comprises optimizing an objective function to train the parameters θ and/or ϕ. In some implementations, the objective function comprises a variational lower bound on a marginal likelihood of the selected distributions. For example, the variational lower bound V may be calculated by substracting the KL distance between the true and approximating posterior distributions from the true log likelihood of the data. The KL term acts as a regularizer on the approximating posterior q_(ϕ)(t|x, y) which, in effect, penalizes q_(ϕ)(t|x, y) for diverging significantly from the prior p_(θ)(t). This may be expressed as follows: V=log p _(θ)(y|x)−KL(q _(ϕ)(t|x,y)∥p _(θ)(t|x,y))

which, assuming independence of t and x given unobserved y, may be reduced to the following formula: V=

_(q) _(ϕ) _((t|x,y))[ log p _(θ)(y|t,x)]−KL(q _(ϕ)(t|x,y)∥p _(θ)(t))

where

_(q) _(ϕ) _((t|x, y)) denotes an expectation value dependent only on the marginal distribution q_(ϕ)(t_(l)|x, y). This can be calculated exactly for certain families of distributions, including the families of distributions provided as examples herein. This expectation value may be expressed as follows:

${{\mathbb{E}}_{q_{\phi}{({{t❘x},y})}}\left\lbrack {\log\;{p_{\theta}\left( {{y❘t},x} \right)}} \right\rbrack} = {{\sum\limits_{l}\;{\left( {1 - {q_{\phi}\left( {{t_{l} = {1❘x}},y} \right)}} \right)\log\;{p_{\theta}\left( {{\hat{y}}_{l} = {y_{l}❘x}} \right)}}} + {{q_{\phi}\left( {{t_{l} = {1❘x}},y} \right)}{\log\left( {1 - {p_{\theta}\left( {{\hat{y}}_{l} = {y_{l}❘x}} \right)}} \right)}}}$

Note that the unary marginal distribution q_(ϕ)(t_(l)=1|x, y) corresponds to the probability that the l^(th) label should be flipped, which is conditional on both the instance x and the observed label y. Accordingly, this model may be viewed intuitively as a tool which uses the approximating posterior distribution to change the l^(th) label if the observed label is incompatible with or unlikely to be combined with the instance x.

The machine learning system may be trained by optimizing the variational lower bound using stochastic gradient descent and/or any other suitable optimization approach. In some implementations, the objective function to be optimized may be expressed as follows:

${\max\limits_{\theta,\phi}{\frac{1}{N}{\sum\limits_{n}^{N}\;{{\mathbb{E}}_{q_{\phi}{({{t❘x^{(n)}},y^{(n)}})}}\left\lbrack {\log\;{p_{\theta}\left( {{y^{(n)}❘t},x^{(n)}} \right)}} \right\rbrack}}}} - {{{KL}\left( {{q_{\phi}\left( {{t❘x^{(n)}},y^{(n)}} \right)} \parallel {p_{\theta}(t)}} \right)}.}$

At 140, the trained machine learning system performs inference based on the trained distributions. In particular, the recognition system proposes (infers) a set of labels ŷ for a given input x based on the conditional distribution p_(θ)(ŷ|x) which it encodes.

The presently-described systems and methods provide a significant degree of flexibility. For instance, the prior distribution may optionally be used to limit the willingness of the instantiated machine learning system to change labels (e.g. by adding one or more spikes and/or by adjusting a mixing parameter, discussed below). The prior distribution may encode a belief about label noise by training the distribution from a “clean” dataset (i.e. one with no or limited errors relative to data encountered in ordinary use) and/or from a noisy dataset (in which case the KL term may contribute to the training of the prior distribution).

At least some implementations described herein accommodate the multi-labeling problem where each instance can have multiple labels. The recognition system selected at 112 has only limited constraints, namely that it accepts data as input and provides classification probabilities as output, although other inputs and/or outputs may be provided. Further, for appropriate pairs of prior and approximating posterior distributions, it may be possible in some circumstances to calculate the objective function analytically and to perform training using widely-available machine learning software, such as Tensorflow™.

FIGS. 3A and 3B show example architectures 300A and 300B, respectively, of the machine learning system instantiated by the presently-described systems and methods. In FIG. 3A, each of recognition system 320 and approximating posterior system 322 receive input data 302 (denoted x), train on that input (and potentially other inputs), and generate parameters. Approximating posterior system 322 also receives input labels 306, denoted y. Approximating posterior system 322 models the approximating posterior distribution; for example, approximating posterior system 322 may comprise a machine learning system, such as a deep neural network, which is configured to provide output based on the selected approximating posterior distribution. Recognition system 320 is also a machine learning system, and may be a type of machine learning system which is the same as or different than approximating posterior system 322. In some implementations, both approximating posterior system 322 and recognition system 320 comprise deep neural networks.

In some implementations, recognition system 320 and approximating posterior system 322 share an input variable transformation system 310, as shown in architecture 300B of FIG. 3B. Transformation system 310 receives input variable 302 (e.g. x) as input and generates a representation 304, denoted f(x), as output. Representation 304 is received by recognition system 320 and approximating posterior system 322 as input. This shared architecture allows for parameter and/or representation sharing between recognition system 320 and approximating posterior system 322, which may reduce the complexity of those systems and/or enable the processing of relatively larger problems than is enabled by architecture 300A in at least some circumstances.

Architectures 300A and 300B define schematically a computing system which has instantiated a machine learning system according to the present disclosure. Aspects of architectures 300A and 300B may be represented in a memory of the computing system, and aspects of architectures may direct the operation of one or more processors of the computing system. For example, shared transformation system 310 may be represented in memory as a deep neural network and its logical structure may direct the operation of one or more processors in the transformation of input 302 to representation 304.

As mentioned above, a variety of prior and approximating posterior distributions may be used in the presently-described systems and methods. The selection of such distributions will impact the calculation of the KL term, among other things. Examples of prior distributions, approximating posterior distributions, and KL term calculations are provided below.

Examples of Prior Distributions

In some implementations, prior distribution p_(θ)(t) comprises a factorial distribution defined by:

${p_{\theta}(t)} = {\prod\limits_{l}\;{\theta_{l}^{t_{l}}\left( {1 - \theta_{l}} \right)}^{({1 - t_{l}})}}$

where θ_(l) is a parameter that controls the frequency with which the posterior distribution flips labels. For example, forcing θ_(l)→0 will penalize the posterior distribution for flipping labels and will therefore result in less frequent label flips. The factorial distribution is relatively simple, which can provide computational advantages, but that simplicity does not allow for the convenient representation of structure or correlations in label noise; in effect, this distribution represents a general belief of the noise level for each label independently.

In some implementations, prior distribution p_(θ)(t) comprises a non-factorial Boltzmann distribution defined by:

${p_{\theta}(t)} = {\frac{1}{Z_{\theta}}e^{{- t^{T}}\theta\; t}}$

where θ is an upper triangular matrix representing the correlations on label changes for different classes and Z_(θ) is a partition function (or normalizing coefficient) which scales p_(θ)(t) to be a valid probability distribution. The Boltzmann distribution permits the representation of structure and is a convenient selection for a variety of posterior distributions.

In some implementations, a Boltzmann prior distribution p_(θ)(t) may be modified to provide a high probability in a particular state via addition of a “spike”. For example, the state t=0 (which corresponds to no flips in the observed label) may be given a higher probability by adding a distribution around that state, such as a delta distribution δ(t). For example, the following distribution may be added:

${p_{\alpha,\theta}(t)} = {{{\left( {1 - \alpha} \right){\delta(t)}} + {\alpha\;{p_{\theta}(t)}\mspace{14mu}{where}\mspace{14mu}{\delta(t)}}} = \left\{ \begin{matrix} 1 & {{{if}\mspace{14mu} t_{l}} = {0\mspace{11mu}{\forall l}}} \\ 0 & {otherwise} \end{matrix} \right.}$

where α is a mixing coefficient which balances the belief between the selected state (0 in this example) and the prior distribution represented by p_(θ)(t). Adding a spike at t=0 will tend to reduce the frequency with which the model changes labels. This effect may be increased by increasing the value of the mixing coefficient α or reduced by decreased the value of α.

Examples of Approximating Posterior Distributions

In some implementations, approximating posterior distribution q_(ϕ)(t|x, y) comprises a factorial distribution defined by:

${q_{\phi}\left( {{t❘x},y} \right)} = {\prod\limits_{l}\;{{\mu_{l}\left( {x,{y;\phi}} \right)}^{t_{l}}\left( {1 - {\mu_{l}\left( {x,{y;\phi}} \right)}} \right)^{({1 - t_{l}})}}}$

where μ_(l)(x, y; ϕ) is a probability distribution with parameters ϕ. For example, μ_(l) may be a sigmoid function generated by an approximating posterior system instantiated as a deep neural network with sigmoid activation in the last layer. The corresponding unary marginal distribution may be calculated analytically as follows: q _(ϕ)(t _(l)=1|x,y)=μ_(l)(x,y;ϕ).

As noted above with respect to the prior distribution, factorial distributions may provide some computational simplicity but tend to come with certain limitations. For instance, since factorial distributions are unimodal, such distributions may perform relatively poorly in circumstances where the true posterior is multi-modal. For example, if the input data x is labelled with “dog”, the states corresponding to no change and switching the label “dog” with “cat” might each have high probability in the true posterior distribution; a factorial distribution may have some degree of difficulty in such circumstances.

In some implementations, approximating posterior distribution q_(ϕ)(t|x, y) comprises a combination of a base distribution q′_(ϕ)(t|x, y) with a spike (e.g. a delta distribution). For example, a spike around the t=0 state may be implemented according to the following formula: q _(ϕ)(t|x,y)=(1−α(x,y))δ(t)+α(x,y)q′ _(ϕ)(t|x,y)

where α(x, y) is a function of x and y which defines a mixing coefficient between the delta and base distributions. For example, α may be a sigmoid function generated by an approximating posterior system instantiated as a deep neural network with sigmoid activation in the last layer. The unary marginal distribution may be calculated according to the following formula: q _(ϕ)(t _(l)=1|x,y)=α(x,y)q′ _(ϕ)(t _(l)=1|x,y).

For example, for a factorial base distribution q′_(ϕ)(t|x, y), a factorial approximating posterior distribution may be implemented according to the following formula:

${q_{\phi}\left( {{t❘x},y} \right)} = {{\left( {1 - {\alpha\left( {x,y} \right)}} \right){\delta(t)}} + {{\alpha\left( {x,y} \right)}{\prod\limits_{l}\;{{\mu_{l}\left( {x,{y;\phi}} \right)}^{t_{l}}\left( {1 - {\mu_{l}\left( {x,{y;\phi}} \right)}} \right)^{({1 - t_{l}})}}}}}$

and the corresponding unary marginal distribution may be calculated according to the following formula: q _(ϕ)(t _(l)=1|x,y)=α(x,y)μ_(l)(x,y;ϕ).

In some circumstances, label flips tend to be highly correlated. For example, in a multi-class problem where each instance belongs to only one class, each change from one label i to another j comprises two flips: one flip to i to turn it “off” and one flip to j to turn it “on”. In the label flip vector t described elsewhere herein, it may then be expected that t_(i) and t_(j) will be highly correlated, since both are set to 1 in such an event.

In some implementations, approximating posterior distribution q_(ϕ)(t|x, y) comprises a directed graphical model. Conceptually, the directed graphical model represents each label as a node and provides the probability of a label flip for each class conditioned on the label flip for the annotated class (i.e. the observed label for the instance). More directly, assuming that an instance x belongs to class i, the probability of a label change for each of the remaining classes may be conditioned on i. For example, the approximating posterior distribution q_(ϕ)(t|x, y) may be defined by:

${q_{\phi}\left( {{t❘x},y} \right)} = {{{q_{\phi}\left( {t_{i}❘x} \right)}{\prod\limits_{j \neq i}\;{q_{\phi}\left( {{t_{j}❘x},t_{i}} \right)}}} = {{\mu_{i}\left( {x;\phi} \right)}^{t_{i}}\left( {1 - {\mu_{i}\left( {x;\phi} \right)}} \right)^{({1 - t_{i}})}{\prod\limits_{j \neq i}\;{{\mu_{j}\left( {x,{t_{i};\phi}} \right)}^{t_{j}}{\left( {1 - {\mu_{j}\left( {x,{t_{i};\phi}} \right)}} \right)^{({1 - t_{j}})}.}}}}}$

The corresponding unary marginal distribution may be calculated according to the following formula:

${q_{\phi}\left( {{t_{l} = {1❘x}},y} \right)} = \left\{ {\begin{matrix} {\mu_{l}\left( {x;\phi} \right)} & {{{if}\mspace{14mu} l} = i} \\ \begin{matrix} {{\left( {1 - {\mu_{i}\left( {x;\phi} \right)}} \right){\mu_{l}\left( {x,{{t_{i} = 0};\phi}} \right)}} +} \\ {{\mu_{i}\left( {x;\phi} \right)}{\mu_{l}\left( {x,{{t_{i} = 1};\phi}} \right)}} \end{matrix} & {otherwise} \end{matrix}.} \right.$ Examples of Determining the KL Term

As noted above, prior and approximating posterior distributions are not necessarily selected in isolation. Determining the KL term in certain acts described herein may depend on characteristics of both the prior and approximating posterior distributions, taken together. In particular, the KL term (sometimes referred to as the “KL divergence”) may be decomposed into two terms:

${{KL}\left\lbrack {{q_{\phi}\left( {{t❘x},y} \right)} \parallel {p_{\phi}(t)}} \right\rbrack} = {{- {\sum\limits_{t}\;{{q_{\phi}\left( {{t❘x},y} \right)}\;\log\mspace{11mu}{p_{\phi}(t)}}}} + {\sum\limits_{t}\;{{q_{\phi}\left( {{t❘x},y} \right)}\log\mspace{11mu}{q_{\phi}\left( {{t❘x},y} \right)}}}}$

where the first sum is sometimes referred to as the “cross-entropy” term and the second sum is sometimes referred to as the “entropy” term. The entropy term depends only on the approximating posterior distribution q_(ϕ)(t|x, y) and so does not vary with the selection of the prior distribution. For example, the entropy of a factorial distribution can be calculated according to the following formula:

${{Entropy}\left( {q_{\phi}\left( {{t❘x},y} \right)} \right)} = {{- {\sum\limits_{l}\;{{\mu_{l}\left( {x,{y;\phi}} \right)}\log\mspace{11mu}{\mu_{l}\left( {x,{y;\phi}} \right)}}}} + {\left( {1 - {\mu_{l}\left( {x,{y;\phi}} \right)}} \right){{\log\left( {1 - {\mu_{l}\left( {x,{y;\phi}} \right)}} \right)}.}}}$

As another example, the entropy of an approximating posterior with a spike defined according to the formula given previously (with a delta distribution and a mixing term) is given by:

Entropy(q_(ϕ)(t❘x, y)) = Entropy(α(x, y)) + α(x, y) Entropy(q_(ϕ)^(′)(t❘x, y)) − α(x, y)q_(ϕ)^(′)(t = 0❘x, y)softplus(−logit  (α(x, y)) − log   q_(ϕ)^(′)(t = 0❘x, y)) − (1 − α(x, y))softplus  (logit  (α(x, y)) + log   q_(ϕ)^(′)(t = 0❘x, y))      where $\mspace{79mu}{{{logit}\mspace{11mu}(u)} = {{\log\frac{u}{1 - u}\mspace{14mu}{and}\mspace{14mu}{softplus}\mspace{11mu}(u)} = {\left( {1 + {\exp(u)}} \right).}}}$

As a further example, the entropy of the example directed graphical model described above may be given by:

${{Entropy}\left( {q_{\phi}\left( {{t❘x},y} \right)} \right)} = {{- {\sum\limits_{t_{i}}\;{{q_{\phi}\left( {t_{i}❘x} \right)}\log\mspace{11mu}{q_{\phi}\left( {t_{i}❘x} \right)}}}} - {\sum\limits_{t_{i}}\;{{q_{\phi}\left( {t_{i}❘x} \right)}{\sum\limits_{j \neq i}\;{{q_{\phi}\left( {{t_{j}❘x},t_{i}} \right)}\log\mspace{11mu}{{q_{\phi}\left( {{t_{j}❘x},t_{i}} \right)}.}}}}}}$

In the case of a Boltzmann prior distribution, the cross-entropy term may be decomposed into a number of unary and pairwise marginal distributions over the approximating posterior distribution as follows:

${{Cross}\mspace{14mu}{{Entropy}\left( {{q_{\phi}\left( {{t❘x},y} \right)} \parallel {p_{\theta}(t)}} \right)}} = {{\sum\limits_{l}\;{h_{l}{q_{\phi}\left( {{t_{l} = {1❘x}},y} \right)}}} + {\sum\limits_{k,l}\;{J_{lk}{q_{\phi}\left( {{t_{k} = 1},{t_{l} = {1❘x}},y} \right)}}} + {\log\mspace{11mu} Z_{\theta}}}$

where the vector h and the matrix J represent respectively the diagonal and upper diagonal part of the matrix θ in a Boltzmann distribution. The value of log Z_(θ) may be computed using annealed importance sampling and/or any other suitable sampling technique. In implementations where the objective function is optimized by gradient descent, the gradient of log Z_(θ) may involve drawing M samples from p_(θ)(t) according to the following formula:

${\frac{\partial}{\partial\theta_{ij}}\log\mspace{11mu} Z_{\theta}} \approx {\frac{1}{M}{\sum\limits_{t^{(m)}\text{∼}{p_{\theta}{(t)}}}\;{{- t_{i}^{(m)}}t_{j}^{(m)}}}}$

where t^((m)) is the m^(th) sample from p_(θ)(t).

The pairwise marginal distribution q_(ϕ)(t_(k)=1, t_(l)=1|x, y) depends on the selection of the approximating posterior distribution. The pairwise marginal distribution q_(ϕ)(t_(k)=1, t_(l)=1|x, y) for a factorial distribution may be expressed as follows: q _(ϕ)(t _(k)=1,t _(l)=1|x,y)=μ_(k)(x,y;ϕ)μ_(l)(x,y;ϕ).

The pairwise marginal distribution q_(ϕ)(t_(k)=1, t_(l)=1|x, y) for a distribution with a spike (if defined as described in the example given above) may be expressed as follows: q _(ϕ)(t _(k)=1,t _(l)=1|x,y)=α(x,y)q′ _(ϕ)(t _(k)=1,t _(l)=1|x,y).

The pairwise marginal distribution q_(ϕ)(t_(k)=1, t_(l)=1|x, y) for a directed graphical model as defined above may be expressed as follows:

${q_{\phi}\left( {{t_{k} = 1},{t_{l} = {1❘x}},y} \right)} = \left\{ {\begin{matrix} {{\mu_{k}\left( {x;\phi} \right)}{\mu_{l}\left( {x,{{t_{i} = 1};\phi}} \right)}} & {{{if}\mspace{14mu} k} = i} \\ {{\mu_{l}\left( {x;\phi} \right)}{\mu_{k}\left( {x,{{t_{i} = 1};\phi}} \right)}} & {{{if}\mspace{14mu} l} = i} \\ \begin{matrix} {{\left( {1 - {\mu_{i}\left( {x;\phi} \right)}} \right){\mu_{k}\left( {x,{{t_{i} = 0};\phi}} \right)}{\mu_{l}\left( {x,{{t_{i} = 0};\phi}} \right)}} +} \\ {{\mu_{i}\left( {x;\phi} \right)}{\mu_{k}\left( {x,{{t_{i} = 1};\phi}} \right)}{\mu_{l}\left( {x,{{t_{i} = 1};\phi}} \right)}} \end{matrix} & {otherwise} \end{matrix}.} \right.$

In the case of a Boltzmann prior distribution with a spike as defined in an example above, the cross-entropy term may be decomposed into a number of unary and pairwise marginal distributions over the approximating posterior distribution as follows:

${{Cross}\mspace{14mu}{{Entropy}\left( {{q_{\phi}\left( {{t❘x},y} \right)} \parallel {p_{\theta}(t)}} \right)}} = {{- {\sum\limits_{t}\;{{q_{\theta}\left( {{t❘x},y} \right)}\log\;{p_{\theta}(t)}}}} - {\log\mspace{11mu}\alpha} - {{q_{\phi}\left( {{t = {0❘x}},y} \right)}{softplus}\mspace{11mu}\left( {{\log\mspace{11mu} Z_{\theta}} - {{logit}\mspace{11mu}(\alpha)}} \right)}}$

where the first term (a summation) is the same as the cross-entropy term for the Boltzmann approximating posterior distribution.

Undirected Graphical Models for Label Noise

In the example directed graphical model of FIG. 2 , true labels ŷ are represented implicitly via the interaction between t and y. In some machine learning models, true labels ŷ are represented explicitly, for example as shown in FIG. 4A. Model 400 a of FIG. 4A is illustrative of a typical approach, in which the conditional dependencies between variables 402(x), 406(y) and 404 (ŷ) are modelled via a directed graphical model (i.e. model 400 a), i.e. a model with directed edges 410.

In the example of model 400 a, true labels ŷ are assumed to be latent during training and, in the multiclass classification problem, may be easily marginalized by enumerating over all possible cases. However, certain challenges exist with such models. For instance, marginalizing true labels ŷ may be intractable (at least with existing methods) in the multi-label classification problem due to the numerous possible configurations of ŷ. As another example, the explaining away phenomenon can present obstacles to making accurate inferences over true labels ŷ.

In some implementations, the conditional dependencies between variables are modelled based on an undirected graph, such as in the example undirected graphical model 400 b of FIG. 4B. Example model 400 b may implement, for example, a conditional random field (CRF) model. In the example representation of model 400 b, true labels ŷ and noisy labels y each conditionally depend on x, represented by edges 420 a and 420 b, respectively. Edge 420 c links ŷ and y, thereby enabling model 400 b to capture correlations between true labels ŷ and noisy labels y. Such correlations may help to infer latent true labels ŷ when only noisy labels y are observed.

In practice, noisy labels may be correlated, resulting in multimodal behavior of the joint probability distribution defined by the model. For example, an image (x) with the true label “cat” (ŷ) may tend to receive noisy labels (y) such as “kitten”, “kitty” and “pet”. Those noisy labels will then tend to be relatively highly correlated with the true “cat” label. However, there may be a weak form of exclusion between “kitten” and “kitty”—that is, a user who uses one such label may be less likely to use another noisy synonym, thereby reducing the probability of synonyms (and thus introducing multimodality). To model such potential multimodal behavior of y, hidden variables h (shown as variable 408) may optionally be introduced to model 400 b. Edge 420 d links noisy labels y to hidden variables h.

Undirected models are flexible and permit certain edges to be omitted while still retaining the ability to model noisy labels. In some implementations, edge 420 b is omitted. In some implementations, edge 420 c is omitted. In some implementations, variable 408 and its associated edges (e.g. edge 420 d) are omitted.

The example CRF model 400 b of FIG. 4B defines a joint probability distribution of y, ŷ, and h conditioned on x, denoted p_(θ)(y, ŷ, h|x). Various conditional distributions may be inferred from the joint distribution, such as p_(θ)(ŷ, h|y, x) or p_(θ)(h|y). The joint distribution uses an energy function E_(θ)(y, ŷ, h, x) that assigns an energy score to the configuration (y, ŷ, h, x) and is parametrized by parameters θ. The joint probability distribution may be defined using a Boltzmann (or Gibbs) distribution as follows:

${p_{\theta}\left( {y,\hat{y},{h❘x}} \right)} = {\frac{1}{Z_{\theta}(x)}{\exp\left( {- {E_{\theta}\left( {y,\hat{y},h,x} \right)}} \right)}}$

where Z_(θ)(x) is the partition function (or normalizing coefficient), which may be defined by:

${Z_{\theta}(x)} = {\sum\limits_{y}\;{\sum\limits_{\hat{y}}\;{\sum\limits_{h}\;{\exp\left( {- {E_{\theta}\left( {y,\hat{y},h,x} \right)}} \right)}}}}$

and the energy function E_(θ)(y, ŷ, h, x) may be defined based on the following quadratic formulation: E _(θ)(y,ŷ,h,x)=−a ^(T) ŷ−b ^(T) y−c ^(T) h−ŷ ^(T) Wy−h ^(T) W′y

where a, b, and c are linear bias terms on ŷ, y, and h, respectively, and W and W′ are pairwise interactions between ŷ and y and between y and h, respectively. Where edges are omitted from the model, the corresponding pairwise term may also be omitted from the energy function. (For example, omitting edge 420 c in model 400 b may result in the omission of pairwise term W from the energy function).

In some implementations, at least one of a, b, c, W, and W′ are functions of the input x. For example, in some implementations, terms a and b are functions of x, whereas c and the pairwise interactions are constants, as shown below: E _(θ)(y,ŷ,h,x)=−a ^(T)(x)ŷ−b ^(T)(x)y−c ^(T) h−ŷ ^(T) Wy−h ^(T) W′y

In some implementations, terms which are functions of x are defined using a machine learning model parametrized by ϕ. The machine learning model may be, for example, a neural network, such as a deep convolutional neural network (CNN) parametrized by ϕ. In such implementations, parameters θ may encapsulate ϕ and any constant terms of E_(θ). For instance, in the example above (where a and b are functions of x), θ may be defined by θ={ϕ, c, W, W′}; this formulation may capture all of the parameters of the CRF model (including the embedded CNN which defines a and b).

In implementations, it may be necessary to calculate further conditional distributions based on the joint distribution p_(θ)(y, ŷ, h|x). For example, in implementations where true labels ŷ are not observed over at least part of the dataset (such as semi-supervised implementations), it may be necessary to consider distributions over true labels ŷ conditioned on observed values (such as noisy labels y and data elements x). Such distributions may take the form of, for example, p_(θ)(ŷ|y, x) and/or p_(θ)(ŷ, h|y, x). In some implementations, conditional distribution p_(θ)(ŷ, h|y, x) takes a factorial distribution, which may be calculated analytically given θ. For instance, it may be calculated based on:

${p_{\theta}\left( {\hat{y},{h❘y},x} \right)} = {\prod\limits_{i}\;{{p_{\theta}\left( {{{\hat{y}}_{i}❘y},x} \right)}{\prod\limits_{j}\;{p_{\theta}\left( {h_{j}❘y} \right)}}}}$

where p_(θ)(ŷ_(i)1|y, x)=σ(a_((i))+W_((i,:))y) and p_(θ)(h_(j)|y)=σ(c_((j))+W′_((j,:))y).

${\sigma(u)} = \frac{1}{\left( {1 + \exp - u} \right)}$ is a sigmoid function and the subscripts (i) and (i,:) indicate the i^(th) element or row in the corresponding vector or matrix, respectively. Other distributions (factorial or otherwise) may alternatively, or additionally, be used for such related conditional distributions. In implementations involving hidden variables h, distributions such as p_(θ)(h|x, y) (which may be written as p_(θ)(h|y) in implementations where h and x are independent, as in model 400 b) may also, or alternatively, be determined by conditioning the joint distribution appropriately. Training Undirected Graphical Models for Label Noise A First Approach

In some implementations, the parameters θ of model 400 b are trained via a semi-supervised approach. A training set of data is annotated with noisy labels (y) and a subset of that training set is provided with “clean” (i.e. true) labels (ŷ). An objective function may be defined based on a combination of the marginal data likelihoods of the clean subset and the noisy set. A suitable method may then be used to learn the parameters of the model, such as the maximum likelihood method.

For example, if D_(N)={(x^((n)), y^((n)))} and D_(C)={(x^((c)), ŷ^((c)), y^((c)))} are disjoint sets representing the noisy-labelled and clean-labelled training data, respectively, then the parameters θ of model 400 b may be trained by maximizing the marginal log likelihood, which may be defined as follows:

${\max\limits_{\theta}{\frac{1}{D_{N}}{\sum\limits_{n}\;{\log\;{p_{\theta}\left( {y^{(n)}❘x^{(n)}} \right)}}}}} + {\frac{1}{D_{C}}{\sum\limits_{c}\;{\log\;{p_{\theta}\left( {y^{(c)},{{\hat{y}}^{(c)}❘x^{(c)}}} \right)}}}}$

where p_(θ)(y^((n))|x^((n)))=Σ_(y,h)p_(θ)(y^((n)), ŷ, h|x^((n))) and p_(θ)(y^((c)), ŷ^((c))|x^((c)))=Σ_(h)p_(θ)(y^((c)), ŷ^((c)), h|x^((c))). Although the presence of hidden variables may pose obstacles to finding an analytically optimized solution, the parameters may be trained using variational inference or other suitable methods. Such methods may involve, for example, maximizing the variational lower bound (also known as the evidence lower bound, or ELBO), which may be formed by subtracting the Kullback-Leibler (KL) divergence between an approximating distribution q and the true joint distribution p_(θ). For instance, the lower bound for the noisy training set's term in the marginal log likelihood (i.e. the first of the two terms in the above marginal log likelihood) may be given by:

θ ⁢ ( x , y ) = ⁢ log ⁢ ⁢ p θ ⁡ ( y ❘ x ) - KL [ q ⁡ ( y ^ , h ❘ y , x ) ∥ p θ ) ⁢ ( y ^ , h ❘ y , x ) ] = ⁢ q ⁡ ( y ^ , h ❘ y , x ) ⁡ [ log ⁢ ⁢ p θ ⁡ ( y , y ^ , h ❘ x ) ] - ⁢ q ⁡ ( y ^ , h ❘ y , x ) ⁡ [ log ⁢ ⁢ q ⁡ ( y ^ , h ❘ y , x ) ]

where q denotes an approximating distribution. In implementations where joint distribution p_(θ) is solvable analytically (e.g. as is the case in a undirected graphical model representing a factorial distribution), q may be set to be equal to p_(θ), thereby avoiding a need to introduce an approximation. In some embodiments, q approximates p_(θ), as described elsewhere herein.

In some implementations, parameters θ are trained using Expectation-Maximization (EM), which involves maximizing over the noisy set D_(N) by iterating between two steps. In the E-step, θ is fixed and the lower bound

_(θ)(x, y) is maximized with respect to approximating distribution q(ŷ, h|y, x). In some implementations, this may be done by setting q(ŷ, h|y, x) to be equal to p_(θ)(ŷ, h|y, x); this may be particularly convenient where p_(θ)(ŷ, h|y, x) is analytic. In the M-step, q is fixed and the lower bound is maximized with respect to model parameters θ. This may be done, for example, by optimizing the

_(q(ŷ, h|y, x))[ log p_(θ)(y, ŷ, h|x)] term using Markov Chain Monte Carlo estimation.

A lower bound

_(θ)(x, y, ŷ) may be defined for the clean training subset D_(C) in a similar way, albeit modified to account for observed true labels ŷ. For example, lower bound

_(θ)(x, y, ŷ) may be given by:

_(θ)(x,y,ŷ)=log p _(θ)(y,ŷ|x)−KL[q(h|y)∥p _(θ)(h|y)]

and may be maximized using similar and/or alternative methods to

_(θ)(x, y, ŷ).

The above semi-supervised approach infers the latent variables using the conditional distribution q(ŷ, h|y, x). However, at the beginning of training when the model parameters are not trained yet, sampling from conditional distribution q(ŷ, h|y, x) may have a relatively low rate of accuracy. This issue is compounded when training a model with strong representation power, such as a CRF model with an embedded CNN, as it may fit to poor conditional distributions early in the training process. In some implementations, this is mitigated by oversampling instances from the clean subset D_(C) and/or weighting the clean log likelihood term

_(θ)(x, y, ŷ) more heavily than the noisy log likelihood term

_(θ)(x, y). Alternatively, or in addition, some implementations provide auxiliary sources of information which can be used to extract information about noisy and true labels early in the training process and/or at other times, for example as described in greater detail below.

Training Undirected Graphical Models for Label Noise Using Auxiliary Distributions

In some implementations, an auxiliary distribution p_(aux) (y, ŷ, h) representing the joint probability of noisy and clean labels, along with some hidden variables h, is used to train parameters in a semi-supervised setting. It is not necessary for the auxiliary distribution to be conditional on input data x. In certain circumstances, such an auxiliary distribution may assist in guiding the model (e.g. model 400 b) to infer true (but latent) labels more accurately than would be provided by the training approach provided in the previous section. The auxiliary distribution may optionally represent information drawn from an explicit auxiliary source of information, such as a knowledge graph and/or from an observed semantic relatedness of labels.

The model (e.g. model 400 b) may be trained based on auxiliary distribution p_(aux)(y, ŷ, h) by, for example, adding a KL distance term to the evidence lower bound which penalizes the model for diverging from the conditional distribution resulting from the auxiliary distribution. Such a modified lower bound may be defined over the noisy set D_(N) as follows:

_(θ) ^(aux)(x,y)=log p _(θ)(y|x)−KL[q(ŷ,h|y,x)∥p _(θ)(ŷ,h|y,x)]−αKL[q(ŷ,h|y,x)∥p _(aux)(ŷ,h|y)

where α is an optional scaling term which may vary over the course of training. In some implementations, α is monotonically decreasing. Scaling term α is non-negative (in the above formulation, at least) and may decrease to 0 or remain greater than 0 throughout training. In some implementations, α is constant. In some implementations, α is not monotonically decreasing and may increase (continuously or non-continuously) during training. It will be appreciated that setting α=0 recovers the original lower bound (i.e.

_(θ) ^(aux)(x, y)=

_(θ)(x, y)) and setting α→∞ causes the model to ignore the original lower bound and be dominated by the effects of the auxiliary distribution.

It may strike some as counterintuitive to introduce an additional term to the lower bound, as additional terms loosen the lower bound (which is generally something to be avoided, as tighter lower bounds are generally considered to provide more accurate results). There are a few observations which justify this action. The first is that adding such a term has been found experimentally by the inventor to improve training results. The second is that adding the additional term regularizes the lower bound—in effect, it provides a benefit (by providing additional information) which outweighs the costs of loosening the bound in at least some circumstances. The third is that, in the case of the noisy lower bound

_(θ) ^(aux)(x, y), the lower bound being loosened is that of the noisy labels (y), whereas in most circumstances an objective of the model is to predict true labels (ŷ), so loosening the lower bound does not necessarily reduce the inferential power of the model with respect to the variables of interest.

In some implementations, hidden variables h are omitted from the additional term—that is, a term of the form αKL[q(ŷ|y, x)∥p_(aux)(ŷ|y) is added. In some implementations, an additional term based on the auxiliary distribution is also added to the lower bound defined over the clean set D_(C), e.g. as follows:

_(θ) ^(aux)(x,y,ŷ)=log p _(θ)(y,ŷ|x)−KL[q(h|y)∥p _(θ)(h|y)]−αKL[q(h|y)∥p _(aux)(h|y).

Auxiliary distribution p_(aux)(y, ŷ, h) may be modelled by an undirected graphical model, such as model 400 c of FIG. 4C. In some implementations, auxiliary joint distribution may be modelled as a Restricted Boltzmann Machine (RBM), which may be similar to CRF model 400 b (although example model 400 c differs from model 400 b at least in that it does not depend on x). Accordingly, auxiliary distribution p_(aux)(y, ŷ, h) may be defined according to the following formula:

${p_{aux}\left( {y,\hat{y},h} \right)} = {\frac{1}{Z_{aux}}{\exp\left( {- {E_{aux}\left( {y,\hat{y},h} \right)}} \right)}}$

where Z_(aux) is the partition function (or normalizing coefficient) defined as above but over the auxiliary energy function E_(aux). The auxiliary energy function E_(aux) may be defined based on the following quadratic formulation: E _(aux)(y,ŷ,h)=−a _(aux) ^(T) ŷ−b _(aux) ^(T) y−c _(aux) ^(T) h−ŷ ^(T) W _(aux) y−h ^(T) W′ _(aux) y

where the parameters of the auxiliary model (i.e. a_(aux), b_(aux), c_(aux), W_(aux), W′_(aux), referred to herein as the “auxiliary parameters”) are trained using a suitable training method, such as by maximizing the log likelihood of y and ŷ. For instance, the auxiliary parameters may be trained via persistent contrastive divergence (PCD) and/or stochastic gradient descent (SGD). The auxiliary parameters may be trained on at least part of the clean subset D_(C) prior to training CRF mode 400 b and may remain fixed during training of CRF model 400 b.

Having defined the auxiliary distribution p_(aux) based on a suitable model, such as model 400 c, the primary inference model (e.g. model 400 b) may be trained based at least in part on the auxiliary distribution. For example, training may involve searching for parameters θ which maximize a lower bound based on the auxiliary distribution (e.g. based on an energy function of the auxiliary distribution). For instance, parameters θ may be trained based on the following example objective function:

max θ ⁢ 1  D N  ⁢ ∑ n ⁢ θ aux ⁢ ( x ( n ) , y ( n ) ) + 1  D C  ⁢ ∑ c ⁢ ⁢ θ aux ⁢ ( x ( c ) , y ( c ) , y ^ ( c ) )

where

_(θ) ^(aux)(x^((n)), y^((n))) and

_(θ) ^(aux)(x^((c)), y^((c)), ŷ^((c))) are the modified lower bounds defined over the noisy and clean data sets, respectively, incorporating additional terms based on the auxiliary distribution as defined above. In some implementations, one of

_(θ) ^(aux)(x^((n)), y^((n))) and

_(θ) ^(aux)(x^((c)), y^((c)), ŷ^((c))) is replaced by the corresponding original lower bound (i.e.

_(θ)(x^((n)), y^((n))) or

_(θ)(x^((c)), y^((c)), ŷ^((c))) respectively). In some embodiments, original lower bounds

_(θ)(x^((n)), y^((n))) and

_(θ)(x^((c)), y^((c)), ŷ^((c))) are used without incorporating an additional term based on the auxiliary distribution and information from the auxiliary distribution is incorporated in the training of parameters θ independently of those terms (e.g. by adding an additional term to the objective function).

Parameters θ may be trained via any suitable method. In some implementations, parameters θ are trained based on Expectation-Maximization (EM), which involves alternating between an E-step and an M-step (e.g. as described above).

In at least some implementations, the E-step involves optimizing an objective function with respect to the approximating distribution q(ŷ, h|y, x) for fixed parameters θ. For example, modified noisy lower bound

_(θ) ^(aux)(x^((n)), y^((n))) may be optimized by solving the following example lower bound optimization function:

${\min\limits_{q}{{KL}\left\lbrack {{q\left( {\hat{y},{h❘y},x} \right)} \parallel {p_{\theta}\left( {\hat{y},{h❘y},x} \right)}} \right\rbrack}} + {\alpha\;{{KL}\left\lbrack {{q\left( {\hat{y},{h❘y},x} \right)} \parallel {p_{aux}\left( {\hat{y},{h❘y}} \right)}} \right\rbrack}}$

which is minimized when the approximating distribution is proportional to an α-weighted geometric mean of the true posterior distribution p_(θ)(ŷ, h|y, x) and the auxiliary distribution p_(aux)(ŷ, h|y, x). In particular, the above example lower bound optimization function is minimized when:

${q\left( {\hat{y},{h❘y},x} \right)} \propto {\left\lbrack {{p_{\theta}\left( {\hat{y},{h❘y},x} \right)}{p_{aux}^{\alpha}\left( {\hat{y},{h❘y}} \right)}} \right\rbrack^{\frac{1}{\alpha + 1}}.}$

In implementations where p_(θ)(ŷ, h|y, x) and p_(aux)(ŷ, h|y) are factorial, the approximating distribution q(ŷ, h|y, x) will also be factorial. For instance, particular values of q may be determinable based on the following formulae:

${q\left( {{{\hat{y}}_{i} = {1❘y}},x} \right)} = {\sigma\left( {\frac{1}{\alpha + 1}\begin{pmatrix} {a_{(i)} + {W_{({i,:})}y} + {\alpha\; a_{{aux}{(i)}}} +} \\ {\alpha\; W_{{aux}{({i,:})}}y} \end{pmatrix}} \right)}$ ${q\left( {h_{j} = {1❘y}} \right)} = {\sigma\left( {\frac{1}{\alpha + 1}\begin{pmatrix} {c_{(j)} + {W_{({j,:})}^{\prime}y} + {\alpha\; c_{{aux}{(j)}}} +} \\ {\alpha\; W_{{aux}{({j,:})}}^{\prime}y} \end{pmatrix}} \right)}$

where a, c, W and W′ are linear and pairwise terms based on the energy function E_(θ) of the joint distribution p_(θ), as described above, and a_(aux), c_(aux), W_(aux) and W′_(aux) are similarly-defined terms based on the energy function E_(aux) of the auxiliary distribution p_(aux).

Similarly, the modified clean lower bound

_(θ) ^(aux)(x, y, ŷ) may be optimized with respect to q(h|y) when:

${q\left( {h❘y} \right)} \propto {\left\lbrack {{p_{\theta}\left( {h❘y} \right)}{p_{\theta}^{aux}\left( {h❘y} \right)}} \right\rbrack^{\frac{1}{\alpha + 1}}.}$

In at least some implementations, the M-step involves optimizing the objective function with respect to θ while q is fixed. In some implementations, the problem of optimizing the objective function is recast as maximizing an expectation of a log-likelihood of the joint distribution, which may be expressed as follows:

max θ ⁢ q ⁡ ( y ^ , h ❘ x , y ) ⁡ [ log ⁢ ⁢ p θ ⁡ ( y , y ^ , h ❘ x ) ] .

This may be done by any suitable optimization technique. In some implementations, the expectation is maximized using a gradient ascent-based technique, which may involve determining gradients across one or both of the modified noisy and clean lower bounds

_(θ) ^(aux) and

_(θ) ^(aux) (and/or their corresponding original lower bounds, as applicable). These gradients may be determined by determining gradients across each of the noisy and clean lower bounds with respect to θ, which in some implementations may be determined as follows:

∂ ∂ θ ⁢ θ aux ⁢ ( x , y ) = ⁢ ∂ ∂ θ ⁢ q ⁡ ( y ^ , h ❘ x , y ) ⁡ [ log ⁢ ⁢ p θ ⁡ ( y , y ^ , h ❘ x ) ] = ⁢ q ⁡ ( y ^ , h ❘ x , y ) ⁡ [ ∂ ∂ θ - E θ ⁡ ( y , y ^ , h , x ) ] - p ⁡ ( y , y ^ , h ❘ x ) ⁢ [ ∂ ∂ θ - E θ ⁡ ( y , y ^ , h , x ) ] ∂ ∂ θ ⁢ θ aux ⁢ ( x , y , y ^ ) = ⁢ ∂ ∂ θ ⁢ q ⁡ ( h ❘ y ) ⁡ [ log ⁢ ⁢ p θ ⁡ ( y , y ^ , h ❘ x ) ] = ⁢ q ⁡ ( h ❘ y ) ⁡ [ ∂ ∂ θ - E θ ⁡ ( y , y ^ , h , x ) ] - p ⁡ ( y , y ^ , h ❘ x ) ⁢ [ ∂ ∂ θ - E θ ⁡ ( y , y ^ , h , x ) ]

where, for each of the above gradients, the first of the summed expectations may be referred to as the “positive phase” and is defined over the approximating distribution q and the second of the summed expectations may be referred to as the “negative phase” and is defined over the joint distribution p_(θ). In some implementations, such as the above example, the negative phases of the noisy and clean gradients are identical, whereas the positive phases differ in that for the clean gradient the true label ŷ is given for each instance so the sampling may only need to be done over the remaining hidden variables (e.g. over h conditioned on y).

The positive phase for each gradient may be determined analytically, by estimation, or by any other suitable technique. In some implementations, the expectation over the joint distribution (i.e. the negative phase) is not determinable analytically and may be determined via estimation, belief propagation, sampling, and/or other non-analytical techniques (referred to generally in the claims appended hereto as determination by approximation). For example, the negative phase may be determined based on Markov Chain Monte Carlo and/or persistent constrastive divergence. (Where PCD is used, a set of particles may be maintained for each training instance and used to seed Markov chains at each iteration of training).

The scaling factor α may be any suitable value during training. For example, α may be a constant value (e.g. provided by a user and/or determined experimentally or analytically). In some implementations, α is initially set to a large value (e.g. values on the order of 10-100 have been found to work well in some experiments, although α is not limited to such values in this disclosure or in the appended claims) and is decreased over the course of training to smaller values. Scaling factor α may reduce to 0 or may remain greater than 0 throughout training. A motivation for this shift in values is that p_(θ)(ŷ, h|y, x) may not be able to accurately predict true labels early in training, but may improve later in training, so the training process may rely more on p_(aux)(ŷ, h|y) early in training and less later in training. Accordingly, scaling factor α may be set approximately inversely proportionately to an expected accuracy of p_(θ)(ŷ, h|y, x) at a given point in training.

Once trained, the trained model may generate predictions of true labels ŷ by determining the value of marginal distribution p_(θ)(ŷ|x) for some input data x. In some implementations, this is done by performing Gibbs sampling over the joint distribution p_(θ)(y, ŷ, h|x) to determine the marginal distribution.

It will be appreciated that particular aspects of the foregoing methods may be implemented by a computing system in various ways. In at least one example method, a CRF model with an embedded CNN is trained by gradient descent based on a noisy dataset D_(N), a clean dataset D_(C), a previously-trained auxiliary distribution p_(aux)(y, ŷ, h), a learning parameter ∈ and a scaling schedule α (which may vary over the course of training, as described above). The method returns model parameters θ={ϕ, a, b, c, W, W′} (although note that terms which are defined by ϕ may not be explicitly represented by θ). An example pseudocode representation of such a method is shown below:

while Stopping criteria is not met do | foreach minibatch {(x^((n)), y^((n))), (x^((c)), ŷ^((c)), y^((c)))} = getMinibatch(D_(N), D_(C)) do |  |Compute q(ŷ, h|y^((n)), x^((n))) for each noisy instance |  |Compute q(h|y^((c))) for each clean instance |  |Do Gibbs sweeps to sample from the current p_(θ)(y, ŷ, h|x^((•))) for each clean/noisy instance |  |(m_(n), m_(c)) ← (# noisy instances in minibatch, # clean instances in minibatch) |  | $\left. \theta\leftarrow{\theta + {ɛ\left( {\frac{1}{m_{n}}{\sum_{n}{\frac{\partial}{\partial\theta}{\mathcal{U}_{\theta}^{aux}\left( {x^{(n)},y^{(n)}} \right)}}}} \right)} + {\frac{1}{m_{c}}{\sum_{c}{\frac{\partial}{\partial\theta}{\mathcal{L}_{\theta}^{aux}\left( {x^{(c)},y^{(c)},{\hat{y}}^{(c)}} \right)}}}}} \right.$ | end end

FIG. 5 shows an example method 500 for instantiating an example machine learning system according to an aspect of the present disclosure. At 510, a joint distribution p_(θ) for modelling label noise is selected—for example, based on the undirected graphical model 400 b of FIG. 4B.

At 512, a representation model is selected. The representation model may be represented by p_(θ)(ŷ|x) and may be selected in any of the ways described above with respect to act 112. For example, the representation model may comprise a CNN modelling various linear and pairwise terms, as described above.

At 514, an approximating distribution is optionally selected. As described above, in some implementations a conditional distribution based on the joint distribution may be used in place of the approximation distribution (or, stated equivalently, the approximating distribution q may be set to be equal to its corresponding analytic distribution p_(θ), so that no approximating is introduced by q).

At 515, an auxiliary distribution is selected. Note that the order of the depicted acts may be varied, so that the auxiliary distribution may be selected, instantiated via its corresponding system (at 519) and/or trained (at 521) prior to the selection, instantiation, and/or training of other models, distributions, and systems (such as the joint distribution and representation model).

One or more of the selections at 510, 512, 514, and 515 may be made by a user and provided to a computing system. Alternatively, or in addition, one or more of the selections at 510, 512, 514, and 515 may be made by the computing system, e.g. in response to identifying a characteristic of a problem, in response to an instruction, and/or by predetermination (e.g. the computing system may only be provided with one option to select from).

At 516, 518, and 519, the representation model, joint distribution, and auxiliary distribution are instantiated by a computing system as a representation system, an inference system, and an auxiliary system, respectively. Instantiation may comprise representing the relevant models and distributions with logical structures in a memory of the computing system. The instantiated systems direct the operation of one or more processors of the computing system. For example, the instantiated systems may comprise deep neural networks which are operable to take the inputs required by the model (or distribution) and to generate outputs which correspond to the outputs of the model (or distribution). Examples of instantiations of acts 518 and 519 are discussed in greater detail above with reference to FIGS. 4B and 4C.

At 530, the machine learning system is trained based on the selected distributions and their associated systems. The auxiliary distribution, joint distribution, and representation model may be trained separately or together. Their training acts are shown as 521, 524, and 526, respectively. In some implementations, the auxiliary distribution is fixed during training of the representation model and joint distribution; it may be trained prior to the training of the machine learning system, may be pre-determined, and/or may be fixed during training of the machine learning system. Accordingly, 521 (corresponding to the training of the auxiliary distribution) is shown as optional by rendering it in dashed lines in FIG. 1 . In some implementations, the representation model and joint distribution are trained together.

At 540, the trained machine learning system performs inference based on the trained joint distribution. In particular, the inference system proposes (infers) a set of true labels ŷ for a given input x based on the conditional distribution p_(θ)(ŷ|x). This may comprise marginalizing other variables (such as y and h), which the computer system may do via, for example, Gibbs sampling or any other suitable marginalization technique.

The above described method(s), process(es), or technique(s) could be implemented by a series of processor readable instructions stored on one or more nontransitory processor-readable media. Where one computing system is described, it will be understood that such computing systems may comprise a plurality of processors (e.g. in communication via a network). The above described method(s), process(es), or technique(s) may include various acts, though those of skill in the art will appreciate that in alternative examples certain acts may be omitted and/or additional acts may be added. Those of skill in the art will appreciate that the illustrated order of the acts is shown for exemplary purposes only and may change in alternative examples. Some of the exemplary acts or operations of the above described method(s), process(es), or technique(s) are performed iteratively. Some acts of the above described method(s), process(es), or technique(s) can be performed during each iteration, after a plurality of iterations, or at the end of all the iterations.

The above description of illustrated implementations, including what is described in the Abstract, is not intended to be exhaustive or to limit the implementations to the precise forms disclosed. Although specific implementations of and examples are described herein for illustrative purposes, various equivalent modifications can be made without departing from the spirit and scope of the disclosure, as will be recognized by those skilled in the relevant art. The teachings provided herein of the various implementations can be applied to other methods of quantum computation, not necessarily the exemplary methods for quantum computation generally described above.

The various implementations described above can be combined to provide further implementations. All of the commonly assigned US patent application publications, US patent applications, foreign patents, and foreign patent applications referred to in this specification and/or listed in the Application Data Sheet are incorporated herein by reference, in their entirety, including but not limited to:

U.S. Patent Application No. 62/322,116;

U.S. Patent Application No. 62/404,591;

U.S. Patent Application No. 62/427,020;

U.S. Patent Application No. 62/508,343;

U.S. Patent Publication No. US2016/047,627;

U.S. patent application Ser. No. 62/427,020;

U.S. Patent Application Ser. No. 15/452,438;

U.S. Patent Publication No. US2016/047,628;

International patent application PCT/US2016/047,628, published as WO 2017/031,357; and

International patent application PCT/US2017/015,401, filed Jan. 27, 2017.

These and other changes can be made to the implementations in light of the above-detailed description. In general, in the following claims, the terms used should not be construed to limit the claims to the specific implementations disclosed in the specification and the claims, but should be construed to include all possible implementations along with the full scope of equivalents to which such claims are entitled. Accordingly, the claims are not limited by the disclosure. 

The invention claimed is:
 1. A method for instantiating a machine learning system for classifying one or more observed elements of an input dataset, the input dataset labelled with one or more observed labels drawn from a plurality of binary labels, the method executed by at least one processor in communication with at least one memory and comprising: instantiating a recognition system in the at least one memory, the recognition system operable to classify elements of the input dataset by generating a classification probability for the at least one observed element; generating a prior distribution, the prior distribution modelling label noise as a set of stochastic label flips on the plurality of binary labels, each label flip indicating a belief in the correctness of a corresponding label; instantiating an approximating posterior system in the at least one memory, the approximating posterior system being a deep neural network operable to generate a posterior probability for one or more label flips for the observed element given the one or more observed labels; modeling an approximating posterior distribution using the approximating posterior system, the approximating posterior distribution approximating a true posterior distribution corresponding to the prior distribution, wherein at least one of the prior and posterior distribution comprises a spike, the spike defining a high-probability state where no labels are flipped; and training at least one of the recognition system and the approximating posterior system based on a training dataset, the prior distribution, and the approximating posterior distribution.
 2. The method of claim 1 wherein the method further comprises instantiating a shared transformation system in the at least one memory, the shared transformation system operable to receive the one or more observed elements and to generate a representation of the observed elements based on one or more shared parameters, wherein the recognition system and the approximating posterior system are operable to receive the representation of the observed elements as input.
 3. The method of claim 2 wherein training at least one of the recognition system and the approximating posterior system comprises training the shared transformation system to generate the one or more shared parameters.
 4. The method of claim 1 further comprising selecting the prior distribution, and selecting the approximating posterior distribution, wherein at least one of selecting the prior distribution, and selecting the approximating posterior distribution comprises selecting a predetermined distribution.
 5. The method of claim 1 wherein the approximating posterior distribution comprises a directed graphical model, the directed graphical model operable to generate, given an initial class, a probability of label change for each of one or more remaining classes.
 6. The method of claim 1 wherein the method further comprises selecting one of the prior distribution and the approximating posterior distribution based on a previous selection of the other one of the prior distribution and the approximating posterior distribution.
 7. The method of claim 6 wherein the prior distribution comprises a Boltzmann distribution with a spike and the approximating posterior distribution comprises a factorial distribution with a spike.
 8. A method for training a machine learning system for classifying one or more observed elements of an input dataset, the input dataset labelled with one or more observed labels, the method executed by at least one processor in communication with at least one memory and comprising: instantiating an inference system in the at least one memory; modelling a joint probability distribution over a plurality of variables using the inference system, the plurality of variables comprising the observed labels and one or more true labels, the joint probability distribution conditional on the input dataset; instantiating an auxiliary system in the at least one memory; modelling an auxiliary probability distribution over the plurality of variables independently of the input dataset using the auxiliary system; instantiating a representation system in the at least one memory, the representation system operable to characterize the one or more interactions between the plurality of variables of the joint distribution; training at least one of the representation system and the inference system based on a first modified lower bound defined over at least a noisy subset of the input dataset, the noisy subset comprising input data and associated observed labels, the first modified lower bound comprising an original lower bound based on the joint probability distribution and an additional term based on the auxiliary probability distribution.
 9. The method of claim 8 wherein training at least one of the representation system and the inference system comprises training at least one of the representation system and the inference system based on a modified clean lower bound defined over a clean subset of the input dataset, the clean subset disjoint from the noisy subset and comprising input data, associated observed labels, and associated true labels, the modified clean lower bound based at least in part on the auxiliary probability distribution.
 10. The method of claim 8 wherein training at least one of the representation system and the inference system comprises determining a gradient over the additional term and training at least one of the representation system and the inference system based on the gradient.
 11. The method of claim 10 wherein determining the gradient over the additional term comprises determining a first gradient over a positive phase of the additional term and a second gradient over a negative phase of the additional term, the first gradient determined analytically and the second gradient determined by approximation.
 12. The method of claim 11 wherein training at least one of the representation system and the inference system comprises optimization of an objective function by expectation maximization.
 13. The method of claim 8 wherein the auxiliary probability distribution is fixed while training at least one of the representation system and the inference system.
 14. The method of claim 13 comprising training the auxiliary probability distribution based on information independent of the training dataset prior to training at least one of the representation system and the inference system.
 15. The method of claim 8 wherein training at least one of the representation system and the inference system comprises training at least one of the representation system and the inference system based on an optimization function comprising a first term based on the joint probability distribution and independent of the auxiliary probability distribution and based on a second term based on the auxiliary probability distribution, wherein at least one of the first and second terms is scaled by a scaling factor.
 16. The method of claim 15 comprising setting the scaling factor to a first value for a first iteration of training and setting the scaling factor to a second value for a second iteration of training.
 17. The method of claim 16 wherein the scaling factor is monotonically decreasing during training.
 18. The method of claim 8 wherein the inference system models the joint probability distribution based on an undirected graphical model, the undirected graphical model comprising one or more undirected edges representing one or more interactions between the plurality of variables of the joint distribution.
 19. The method of claim 18 wherein the inference system models the auxiliary probability distribution based on an auxiliary undirected graphical model, the auxiliary undirected graphical model comprising an undirected subgraph of the undirected graphical model of the inference system. 